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<div style="margin-left: 40px;"><b>3. <a name="." href="#Embedded_Controllers_">
	Embedded Controllers</a><a name=".">.</a></b><br>
<div style="margin-left: 40px;">3.1 <a href="#Furutas_pendulum">Furuta's pendulum</a><br>
3.2. <a href="#Cartpole">Cartpole</a><br>
3.3. <a href="#Acrobot_and_Pendubot_">Acrobot and Pendubot</a><br>
3.4. <a href="#Inertia_Wheel">Inertia Wheel</a><br>
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	&nbsp;</div>
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	<p><b><a name="Embedded_Controllers_">3. Embedded Controllers</a></b></p>
	<p align="justify">VISUNS incorporates a library with some controllers 
	embedded for each underactuated system. The aim of this section consists in 
	the explanation about the control strategy used in the development of the 
	system's controllers. These controllers are named <i>Default,</i> and the 
	user may use them in order to proof the system.</p>
	<p align="justify"><span style="font-family: Times New Roman">The main 
	control methods applied to examples Furuta, Cartpole, Pendubot and Acrobot 
	are based on swing-up of the pendulum from its downward position and then 
	switching to a balancing controller that is designed using a linearization 
	technique or gain scheduling to balance the pendulum. </span></p>
	<p align="right"><i><a href="#.">Back Top</a></i></p>
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	<blockquote>
		<p align="justify"><b>3.1 <a name="Furutas_pendulum">Furuta's pendulum</a></b></p>
		<p align="justify">
		<img border="0" src="images/furucontrol.JPG" width="882" height="423"></p>
		<p align="justify">
		<img border="0" src="images/furucontrol1.JPG" width="875" height="466"></p>
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	<blockquote>
		<p align="justify">
		<img border="0" src="images/furucontrol2.JPG" width="879" height="388"></p>
		<p align="justify">&nbsp;Simulating the controller using VISUNS, the 
		response of the system, using RungeKutta numerical integrator with a 
		step time of 0.001s and total<br>
	&nbsp;time of 20s, yields:</p>
		<p align="justify">
		<img border="0" src="images/furucontrol3.JPG" width="878" height="168"></p>
		<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
			<p align="justify"><b>3.2. <a name="Cartpole">Cartpole</a></b></p>
		<p align="justify">Remembering the nonlinear equations:</p>
		<p align="justify">
		<img border="0" src="images/cartdyn2.JPG" width="448" height="272"></p>
			<p><span style="line-height: 115%; font-family: Times New Roman">The 
			Motor Dynamics equations included:</span></p>
			<p>
			<img border="0" src="images/cartdyn4.JPG" width="185" height="56"></p>
			<p align="justify">The same approach used in Furuta's pendulum was 
			applied for the linearization of the system. The linear model 
			includes the DC-motor dynamics:</p>
		<p align="justify">
		<img border="0" src="images/cartcontrol.JPG" width="516" height="303"></p>
		<p align="justify">
		<img border="0" src="images/cartcontrol1.JPG" width="876" height="297"></p>
		<p align="justify">Simulating the controller using VISUNS, the response 
		of the system, using RungeKutta numerical integrator with a step time of 
		0.001s and total<br>
		time of 20s, yields:</p>
		<p align="justify">
		<img border="0" src="images/cartcontrol2.JPG" width="864" height="178"></p>
		<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
			<p align="justify"><b><a name="Acrobot_and_Pendubot_">3.3. Acrobot 
			and Pendubot</a></b></p>
		<p align="justify">From The non-linear model:</p>
		<p align="justify">
			<img border="0" src="images/acropendydyn.JPG" width="864" height="379"></p>
		<p align="justify">
		<img border="0" src="images/controlpen1.jpg" width="872" height="499"></p>
		<p align="justify">
		<img border="0" src="images/controlpen2.jpg" width="869" height="480"></p>
		<p align="justify">
		<img border="0" src="images/controlpen3.jpg" width="871" height="188"></p>
		<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
			<p align="justify"><b><a name="Inertia_Wheel">3.4. Inertia Wheel</a></b></p>
		<p align="justify">
		<img border="0" src="images/wheelcontrol.JPG" width="872" height="519"></p>
		<p align="justify">
		<img border="0" src="images/wheelcontrol2.JPG" width="849" height="545"><img border="0" src="images/wheelcontrol3.JPG" width="881" height="314"></p>
		<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
			<p align="justify">&nbsp;</p>
		<p align="justify">&nbsp;</p>
	</blockquote>
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